WATER RESOURCES RESEARCH, VOL. 19, NO. 5, PAGES 1266-1270, OCTOBER 1983 Control of DO Level in a River Under Uncertainty R. C. SPEAR Department of Biomedical and Environmental Health Sciences, School of Public Health University of California, Berkeley G. M. HORNBERGER Department of Environmental Sciences, University of Virginia A previously developed regionalized sensitivity analysis for exposing critical uncertainties in models of environmental systems is extended to study control of systems for which there is a good dealof uncer- tainty in the mathematical modelused to describe the appropriate physical, chemical, and biological processes. The method is based on a binary classification of Monte Carlo simulation results as being eithersatisfactory or unsatisfactory in terms of controller performance. Contrasts in parameters associ- ated with the two classes are elucidated by statistical analysis. This allows the selection of a setof control parameters that maximizes the probability of acceptable behavior in the presence of uncertainty in process parameters. The method is applied to the problem of regulating the discharge from a lagoon with the object of preventing DO from falling below a predetermined standard. It was found that for this system the desired behavior of the controlled process canbe achieved with a probability of 0.84 with a particularly simple controller design. Nevertheless, the results suggest that evenmodest levels of uncer- tainty in the process parameters can have a considerable effect on the controller performance and that additional attention should bedevoted to thedesign of robust controllers for environmental systems. INTRODUCTION Over the past 10 or 15 years there has been a sustained interest in the application of control theory to water quality problems in river systems. Many of these investigations have focused on DO/BOD dynamics and have employed either dis- tributed [Tarassov et al., 1969] or lumped parameter models of one sort or another [Kendrick et al., 1970; Young and Beck, 1974; Ozunger and Perkins, 1979]. The control methodologies applied to these models have included dynamic programing [Naito et al., 1972], duality theory [Varaiya, 1972], differ- ential game theory [Ozunger and Perkins, 1979], procedures based on pole placement [Young and Beck,1974; Gourishank- ar and Raman, 1977], forms of hierarchical control [Tarnura, 1974] and Monte Carlo methods [Whitehead and Young, 1979]. Singh[1975] pointed out that many of these approaches are characterized by the considerable computational burden re- quired to implement the control scheme. He proposed a sub- optimal control scheme with more modestcomputationalre- quirements to deal with this practical problem,but the fact remains that much of the reportedwork is of principally theo- retical interest.An exceptionis the work of Young and Beck [1974], who carried out field studiesaimed at giving some insightinto the adequacy of their modelingapproachfor con- trol purposes. Their model was subsequent13/used, in a sim- plified form, for theoreticalstudies by Singh[1975], Ozunger and Perkins [1979], and Gourishankarand Raman [1977]. However, none of these authors included the "sustained sun- light" term that Young and Beck found necessary to account for photosynthetic activity in the river and which their data suggested to be important to the overall DO/BOD dynamics. This omission led us to speculate on the effectof uncertainty in model structure or in process parameter values on the design and/or operation of the rather elaborate control schemes that have beendeveloped. In systems with major bio- Copyright by the American Geophysical Union. Paper number 3W1224. 0043-1397/83/003 W- 1224505.00 logical components, uncertainty must be regarded as the rule rather than the exception. To investigate this issue in the con- text of the DO/BOD problem,we chose to studythe impli- cationsof parametric uncertainty on the rather practical ap- proach to the single-reach control problem taken by Young and Beck[1974]. We carriedtheir rejection of the optimal control approach one step further by assuming that the important practical issue was simply to keep the DO concentration in the reach above some minimum level. The issue,then, was to determine the likelihood that the controller would be able to maintain an acceptable DO level in the presence of significant para- metric uncertainty and, furthermore, to identify the key sources of uncertainty that affectcontroller performance. We havepreviously developed a regional sensitivity analysis procedure for exposing the critical uncertainties in modelsof environmental systems [Spearand Hornberger, 1980;Hornber- ger and Spear, 1981]. This procedure depends upon an ability to construct plausible model structures, to estimate broad rangesof parametervaluesfrom limited field data or from the literature, and to define,rather loosely,the system behavior that is associated with the environmental problem(e.g.,see Hornbergerand Spear [1980]). The last of these, the behav- ioral definition, is crucial to the method, and it is worth em- phasizing that the defining algorithmneed not be analytic' thresholds, topological conditions, logicalconditions, etc. are all permissible. The essential features of our sensitivity analysis procedure are based on the following assumptions. 1. The problemunder investigation can be qualitatively characterized by specific patterns of system response that define the "behavior" of concern. 2. One or moremathematical models of the system can be developed based on the relevant physical, chemical, or biologi- cal mechanisms that are assumed to underlie the problem behavior. 3. These models can be parameterized by statistical distri- butions rather than point estimates as a means of incorpor- atingthe uncertainty in the "actual" values of the parameters. If, in a particular case, these conditions can be met, it is 1266